Electrochemical machining (ECM) is a process in which an electrically conductive metal workpiece is shaped by removing material through anodic dissolution. In ECM, the workpiece comprises the anode in an electrolytic cell, the tool comprises the cathode, and an electrolyte is pumped through a gap between the workpiece and the tool. When a potential difference is applied between the tool and the workpiece, current flows through the electrolyte as a result of electrochemical reactions taking place at the surfaces of both electrodes. The reaction at the anode workpiece surface removes material by the oxidation of metal atoms, while the cathode surface is typically unaffected by the reduction reaction occurring there. ECM is a non-contact machining process that can quickly shape any electrically conductive material regardless of the hardness or toughness of the material. The ECM process is also advantageous because it does not produce residual stresses in the workpiece.
In ECM, the tool is formed as an approximately complementary shape of the desired workpiece shape. The tool geometry is copied into the workpiece by electrochemical dissolution to obtain the desired workpiece shape. During the copying process, a gap is established between the tool and the workpiece, as the tool feed rate becomes substantially equal to the rate of workpiece dissolution. The gap size often varies significantly at different locations over the machining region due to, primarily, the non-uniformity of the electric field and temperature field in the region. As a result, the workpiece does not take on the exact shape of the tool. To obtain a desired workpiece shape with desired accuracy, the tool geometry is modified to compensate for the non-uniform gap distribution over the machining region.
Modeling of the relationship between the shape of the tool and the resulting shape of the workpiece can be complex due to modeling of the electrolyte flow, modeling of the electrochemical reactions in the gap region, tool surface, and workpiece surface, and modeling of the electric field which is influenced by all of these factors. Thus, the problem of predicting the workpiece surface resulting from a known tool shape (commonly known as xe2x80x9cthe direct problemxe2x80x9d) can be relatively complex.
The problem of determining a tool shape based on a desired workpiece shape (commonly known as xe2x80x9cthe inverse problemxe2x80x9d), introduces an additional degree of complexity. Known methods of determining a tool shape based on a desired workpiece shape typically utilize an iterative process. For example, Shuvra Das et al., Use of Boundary Element Method for the Determination of Tool Shape in Electrochemical Machining, International Journal for Numerical Methods in Engineering, Vol. 35, 1045-1054 (1992), describes an algorithm based on the boundary integral equation technique which utilizes a non-linear optimization method. Through an iterative process, the shape of the cathode tool is determined by minimizing a functional. In this approach, the inverse problem is converted to a direct problem by assuming a tool shape and its boundary conditions, and the shape is adjusted based on the feedback from the computed results. This approach typically takes a large amount of computational resources to complete the iterations and may have problems with convergence. The iterations also introduce approximations which adversely affect the accuracy of the results.
It would be desirable, therefore, to have a method for directly determining the tool shape needed to produce a desired workpiece shape, without iteration, and without approximation.
A method of shaping a tool, according to an exemplary embodiment of the invention, comprises the steps of defining a shape of an article to be formed with the tool as a plurality of first elements; defining an initial shape of the tool as a plurality of second elements; determining an electric potential of each of the first and second elements; determining an equipotential line between the article and the initial tool shape based on the electric potential of the first and second elements; and forming the tool to have a shape coincident with the equipotential line.